What could we really do with a very powerful particle collider?

A particle collider is a large and complex facility that collides a single beam of particles against a stationary target or two beams head-on usually at very fast speeds. They are usually very expensive and require years of designing and building. Examples of colliders include the Large Hadron Collider near Geneva, Switzerland and Relativistic Heavy Ion Collider in New York, USA.

Future colliders may be bigger and more powerful than the current ones and may be used to know the following:

1. Whether supersymmetry is realized in nature which will also imply that all known particles have supersymmetric partners. 
2. If extra dimensions exist and how we can detect them.

   

   Some theories attempt to unify the fundamental forces by adding extra dimensions.

    
3. What exactly dark matter is. 

   

   Dark Matter

4. Whether the masses of elementary particles are generated by the Higgs mechanism via electroweak symmetry breaking. 
5. Why gravity is many times weaker than the other 3 fundamental forces (electromagnetism, strong interaction, and weak interaction)

   

   Gravity bending spacetime

6. Whether the electroweak force and the strong nuclear force are different manifestations of one universal unified force. 
7. Whether there are other additional sources of quark flavour mixing, apart from those already present within the Standard Model. 
8. Why there are violations of the symmetry between matter and antimatter. 
9. What the exact nature of quark-gluon plasma (quark soup) is. This is a state of matter in quantum chromodynamics (QCD) which is hypothesized to exist at extremely high temperature, density, or both temperature and density.

Is String Theory Falsifiable?

Yes

String theory is a physics framework whereby objects called strings are the most fundamental objects in the universe as opposed to particles in particle physics. The strings are one dimensional and are really small (size of 1 Planck length). To show how small these strings are, imagine if a particle or dot about 0.1 mm in size (which is approximately the smallest the unaided human eye can see) were magnified in size to be as large as the observable universe, then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.1 mm dot. It's probably as small as things get.

So how can string theory be proven:

1. Experimental evidence confirming supersymmetry. One of the predictions of string theory is that at high energies we should start to see evidence of a symmetry that gives every particle that transmits a force, a partner particle. This is called supersymmetry. The partner particles are called superpartners. For string theory to be consistent, supersymmetry seems to be required at some level. Since supersymmetry is a required component of string theory, any discovered supersymmetry would be consistent with string theory. 
2. Evidence of extra dimensions. Superstring theory for example requires 10 spacetime dimensions. If extra dimensions exist, it's possible they must be hidden from us by some physical mechanism. Properties of extra dimensions can by set by particle experiments such as those at the Large Hadron Collider. 
3. String harmonics. Every string, in theory, has a unique resonance, or harmonic. Different harmonics determine different fundamental particles. String harmonics could be experimentally tested in a particle collider. However the collider will need to be about a thousand trillion times more powerful than the Large Hadron Collider. So it might be a few years before we can experimentally verify this. 
4. AdS/CFT correspondence. The anti-de Sitter/conformal field theory correspondence, is a conjectured relationship between two kinds of physical theories. It is a relationship which says that string theory is in certain cases equivalent to a quantum field theory. The duality represents a major advance in our understanding of string theory and quantum gravity. It provides a non-perturbative formulation of string theory with certain boundary conditions and it is also the most successful realization of the holographic principle. Quantum gravity is the branch of physics that seeks to describe gravity using the principles of quantum mechanics. Currently, the most popular approach to quantum gravity is string theory. AdS/CFT is expected to make quantitative and qualitative predictions and this can be known using data from the Large Hadron Collider. 
5. Bubble nucleation. In the theoretical physics of the false vacuum, this is when the system moves to a lower energy state – either the true vacuum, or another, lower energy vacuum. Many phases in string theory have very large, positive vacuum energy. Regions of the universe that are in such a phase will inflate exponentially rapidly in a process known as eternal inflation. As such, the theory predicts that most of the universe is very rapidly expanding. Some properties of the Higgs boson, for example, could be used to determine the possibility of vacuum collapse.